Why $T_{A}M(n) = M(n)$ and why $T_{f(A)}S(n) = S(n)$ and why $M(n)$ & $S(n)$ are manifolds. (Guillemin & Pollack p.23)

Question

Why $T_{A}M(n) = M(n)$ and why $T_{f(A)}S(n) = S(n)$ and why $M(n)$ & $S(n)$ are manifolds?

$M(n)$ is the space of all $n x n$ matrices and $S(n)$ is the space of all $n x n$ symmetric matrices.

A manifold definition is: a subset $X$ of some ambient euclidean space $R^{N}$ is a k-dimensional manifold if it is locally diffeomorphic to $R^k$.

Could anyone clarify the above questions for me please?

Answer 1

If you have a vector space $V$, for any element $v\in V$, we always have $T_vV = V$. The space $M(n)$ of matrices is a vector space, and $S(n)$ is a vector subspace.